1. Field of the Invention
The present invention relates to a method for quickly forming a stochastic model representative of the distribution of a physical quantity such as the permeability, for example, in a porous heterogeneous medium, which is calibrated in relation to dynamic data, by suitable selection of linearly combined geostatistical realizations.
2. Description of the Prior Art
Optimization in a stochastic context determines realizations of a stochastic model which satisfy a set of data observed in the field. In reservoir engineering, the realizations to be identified correspond to representations, in the reservoir field, of the distribution of carrying properties such as the permeability or the porosity. These realizations form numerical reservoir models. The available data are, for example, punctual permeability or porosity measurements, a spatial variability model determined according to punctual measurements or data directly related to the fluid flows in an underground reservoir, that is pressures, breakthrough times, flow rates, etc. The latter are often non linearly related to the physical properties to be modelled. A randomly drawn realization is generally not in accordance with the whole of the data collected. Coherence in relation to the data is ensured in the model by means of an inverse procedure.
Studies integrated in reservoir engineering mainly have two objectives:                On mature fields, the reservoir engineer tries to quantify the uncertainties linked with the production forecasts;        On projects under development or entering into a new production stage, the reservoir engineer wants to be able to test various production scenarios in order to make risk studies.        
In this context, the use of geostatistics as estimation methods as well as stochastic simulation methods has become common practice. Current geostatistical simulation tools allow quickly generating oil reservoir models containing several million grid cells. The challenges linked with the use of such models are mainly of two orders:
[1] On the one hand, it must be possible to integrate the available data to update the reservoir model while ensuring conservation of the geostatistical properties of the initial geological model;
[2] On the other hand, the inverse problem associated with this data integration has to be solved within time limits compatible with the economic constraints.
In both cases, parameterization of the geological model plays an essential part. A conventional approach reduces the number of parameters and accounts only for those having a maximum sensitivity. The pilot point method initially presented by:                Ramarao, B. S., LaVenue, A. M., de Marsily, G. & Marietta, M. G., “Pilot Point Methodology for Automated Calibration of an Ensemble of Conditionally Simulated        
Transmissivity Fields: 1. Theory and Computational Experiments”, Water Resources Research, 31(3): 475-493, Mar. 1995 allows performing a historical matching on a reservoir model parameterized by the well data and a certain number of pilot points specified by the user. Updating of the model is however located in the neighborhood of the pilot points.
The method of zoning the reservoir into main units or zones having constant petrophysical properties has been initially presented by:                Bissel, R., “Calculating Optimal Parameters for History Matching”, Proceedings of the 4th European Conference on the Mathematics of Oil Recovery (ECMOR IV), 1994.        
It allows historical matching to be performed as long as the zonation selected is correct from a geological point of view. Point [1] is however not respected.
Another approach lies in the multiscale parameterization for which the problem is solved successively on increasingly fine scales. It has the drawback of generally leading to an overparameterization because all the degrees of freedom of the lower scale are used, whereas only some of them would be necessary to explain the data.
Adaptive multiscale approaches allow this drawback to be corrected. The concept of refinement indicator presented by:                Chavent, G., & Bissell, R., “Indicator for the Refinement of Parameterization”, Proceedings of the International Symposium on Inverse Problems in Engineering Mechanics, Nagano, Japan, p. 185-190, 1998allows identification of the degrees of freedom useful for explaining the data while avoiding the trap of overparameterization.        
More recently, a geostatistical parameterization technique has been introduced to constrain, by gradual deformation, the stochastic realizations to data on which they depend non-linearly. It is the object of French patents 2,780,798 and 2,795,841 filed by the assignee. It is also described in the following publication:                Roggero, F., & Hu, L. Y., (1998), “Gradual Deformation of Continuous Geostatistical Models for History Matching”, SPE 49004.        
It allows performing history matching while keeping the initial geostatistical properties of the reservoir model. Parameterization then amounts to the gradual deformation parameters from which the user calculates the coefficients of the linear combination. The geological model being parameterized by a linear combination of geostatistical realizations, point [1] can be satisfied by a specific constraint on the coefficients of this linear combination.
Point [2] involves two other conditions from the moment that the gradual deformation method is used. The issue is to know:    [3] Which number N of geostatistical realizations is to be considered for the linear combination, and    [4] How to select as efficiently as possible the N optimum geostatistical realizations (fastest decrease of the objective function of the inverse problem considered).
A conventional approach concerning point [3] carries out a decomposition into eigenvalues and eigenvectors. The different eigenvalues obtained allow finding a compromise between the uncertainty obtained on the parameters at the end of the calibration and the number of parameters that can be estimated from the available data.
According to another approach, still within the context of a parameterization using the gradual deformation method, it is considered that any geostatistical realization still is a contribution, even if it is minimal, to the decrease of objective function (J). Consequently, a certain number of optimum geostatistical realizations are linearly combined by means of the gradual deformation method. These optimum realizations themselves result from a linear combination of initial geostatistical realizations whose combination coefficients are so selected as to provide a gradual deformation search direction that is as close as possible to the direction of descent given by the gradients. This approach forms the object of French patent application 02/13,632 filed by the assignee.
Concerning point [4], no approach has been proposed to date allowing making an a priori selection of the realizations (or maps) used in the linear combination within the context of history matching. Only a technique allowing an a priori selection of the geostatistical realizations corresponding to the extreme production scenarios within an uncertainty quantification context has been proposed by:                Roggero, F., “Direct Selection of Stochastic Model Realizations Contrained to Historical Data”, SPE 38731, 1997.        